Global scaling of precipitation extremes using near-surface air temperature and dew point temperature

Global warming has altered the energy budget and water cycle processes of the land–atmosphere system, which has resulted in significant effects on precipitation extremes. Previous studies have identified a hook structure between near-surface temperature and precipitation extremes, in which extremes increase with temperature rises and decline thereafter. However, the underlying physical mechanisms of this association remain poorly understood. In this study, global-scale responses of precipitation extremes to near-surface air temperature (SAT) and dew point temperature (DPT) were quantified using the ERA5 reanalysis dataset. The results reveal a hook structure between precipitation extremes scaling and temperature, for both SAT and DPT, over many regions worldwide. The peak point temperature (T pp) ranges from 15 °C to 25 °C, increasing as latitude decreased. The association of precipitation extremes with SAT is negative in many areas in the tropics, whereas that with DPT is almost always positive; this suggests that moisture supply is the main factor limiting precipitation at higher surface temperatures. The hook structure and scaling rates incompatible with Clausius–Clapeyron scaling are associated with various factors including precipitation duration, total column water vapour, convective available potential energy, and relative humidity.


Introduction
Anthropogenic warming due to increasing greenhouse gas emissions has altered the terrestrial energy budget and water cycle (Erler et al 2019, Arsenault et al 2020, Yin et al 2023a). These changes have exacerbated the uneven distribution of water resources, resulting in more severe weather-related disasters (Broderick et al 2019, Do et al 2020, Yin et al 2022.
During the past few decades, storms and floods caused by precipitation extremes have resulted in economic losses of more than USD 30 billion per year (Roxy et al 2017), and are thus among the most pressing societal issues worldwide. Therefore, understanding the association between precipitation extremes and temperature is vital for projecting precipitation intensity under future warming (Bui et al 2019). Precipitation extremes are typically assumed to be constrained by the amount of precipitable water in the atmosphere. As the Clausius-Clapeyron (C-C) relationship suggests, saturation water vapour pressure monotonically increases with warming temperature and scales at approximately 6.8% • C −1 (Berg et al 2013, Hegerl et al 2015, Gao et al 2018. Thus, rising temperatures are expected to lead to more precipitation extremes due to the increasing water vapour holding capacity of the atmosphere (Trenberth et al 2003, O'Gorman 2015. During the past decade, C-C scaling (∼6.8% • C −1 ) has been widely used to assess the effects of global warming on precipitation extremes (Allen and Ingram 2002, O'Gorman and Muller 2010). Numerous studies have examined the relationship between precipitation extremes and temperature. They have found that scaling rates range from the super C-C to sub-C-C or are negative (Drobinski et al 2016, Formayer andFritz 2017). Some studies have found that precipitation extremes increase at medium to low surface temperatures and decrease at higher surface temperatures (Drobinski et al 2016, Yin et al 2018, especially in lower latitudes (Sullivan et al 2020, Zeder andFischer 2020). This hook structure indicates that the relationship between precipitation extremes and temperature deviates from that expected C-C scaling (Pfahl et al 2017, Yin et al 2021; thus, the robustness of projections of precipitation extremes based on the precipitationtemperature (P-T) relationship is reduced under a warming climate (Zhang et al 2017, Sun et al 2020. Model simulations show that the scaling rate of future extremes are increasing faster than those obtained from observational scaling (Bao et al 2017), suggesting a mismatch of observations and projections in evaluating the P-T relationship.
The underlying mechanism behind substantial deviations of the precipitation-temperature relationship from C-C scaling are widely discussed. Lenderink and van Meijgaard (2010) inferred that convective precipitation is the primary driver for the super C-C rate at higher temperature. While other studies further classified the convective and stratified precipitation, and found that the super C-C rate is related to the transition between stratified and convective events (Haerter andBerg 2009, Berg andHaerter 2013). It is widely reported that scaling rates differ between regions, precipitation durations and seasons (Berg et al 2009, 2013, Panthou et al 2014, Ali et al 2018, Yin et al 2018. Some studies reported higher scaling rates for hourly or sub-hourly precipitation than for daily precipitation, and highlight the important role of wet event duration in constraining the scaling rates (Lenderink and Van Meijgaard 2008, Utsumi et al 2011). Visser et al (2021 showed an improved consistency of scaling results through precipitation event-based analyses taking account of duration, suggesting that the divergent scaling rate is also related to a coarser daily-based analyses. Some studies attribute the negative scale and hook structure to cooling effects, including evaporative cooling and cold air movement after rainfall events (Ali and Mishra 2017, Bao et al 2017, Gao et al 2020. However, the notion that cooling effects lead to negative scaling rates is considered controversial. Barbero et al (2018) argued that cooling effect shown in Bao et al (2017) is related to the temporal autocorrelation of temperature sampling. In addition, the P-T relation is more consistent with C-C scaling when considering the temperature that represents the atmospheric condition during storms (Lenderink and van Meijgaard 2010, Visser et al 2020). As the maximum relative humidity (RH) decreases abruptly at a temperatures above ∼26 • C, restricted moisture availability appears to control precipitation extremes at higher temperatures (Berg et al 2009, Hardwick Jones et al 2010, Roderick et al 2019. Therefore, some studies have recommended using DPT instead of near-surface air temperature (SAT) as a temperature proxy (Ali and Mishra 2017, Wasko et al 2018, Bui et al 2019. Extreme daily precipitation scaling results based on DPT are more consistent with the expected C-C rate than those based on SAT (Ali et al 2018, Zhang et al 2019b), because SAT is correlated with moisture availability and DPT is less responsive to storm duration (Wasko et al 2018) and precipitation type (Bui et al 2019). However, some studies have also proposed that using atmospheric DPT to investigate the sensitivity of precipitation extremes to temperature offers little advantage over surface temperature (Bui et al 2019). Wang and Sun (2022) introduced the variable of atmospheric saturation deficit (SD) in the analysis of extreme precipitation and found that when controlled for SD, precipitation extremes scale positively.
Precipitation extremes usually indicate sufficient water vapour and an unstable atmosphere (Pfahl et al 2017), involving complex interactions between thermal and dynamic processes. Previous studies have focused on the influence of saturated water vapour pressure on severe weather; while analyses of the influence of climate factors on precipitation are lacking in comparison. For example, the impacts of vertical wind speed, total column water vapour (TCWV), and convective available potential energy (CAPE) on precipitation extremes are rarely addressed.
The objectives of this study were to determine whether DPT addresses the inconsistency of the sensitivity of precipitation extremes under global warming, investigate how rainfall duration and geography influence the scaling of the relationship between precipitation and temperature, and the impact of climate factors such as CAPE and TCWV on precipitation processes. To address these research questions, we analysed the ERA5 (ECMWF Reanalysis v5) dataset in terms of the associations of hourly and daily precipitation extremes with DPT and SAT. Then, we divided hourly precipitation data into 3-and 6-hourly, and daily, precipitation intensity subsets and calculated the scaling rate for each subset. We also determined the influence of other factors on extreme precipitation events, including RH, CAPE, and TCWV, in association with DPT.

Study area and data
The ERA5 product is produced by the European Centre for Medium-range Weather Forecasts, which provides real-time global hourly meteorological data. ERA5 data are produced by combining model simulations based on the laws of physics with observations; data assimilation is performed using the integrated forecasting system (Cy31r2) (Hersbach et al 2018). Many regions of the world lack of observational data with continuous spatiotemporal scales (Wasko et al 2016). The advent of reanalysis products allows for extended analyses of ungauged regions. Numerous studies have used the ERA5 product in evaluating the P-T relationship, and found that the dataset can well capture the characteristic of P-T scaling in reference with the observational data (Moustakis et al 2020, Aleshina et al 2021, Ali et al 2021. We accumulated ERA5 global hourly total precipitation, 2 m SAT, 2 m DPT, TCWV, CAPE, and 750 hPa RH (%) data with a spatial resolution of 1.0 • × 1.0 • for the period 1979-2020 into 3-hourly, 6-hourly and daily data. Near-surface RH was calculated from SAT and DPT. As scaling rates are usually dependent on the climate zone (Panthou et al 2014), we roughly divided the globe into five climate zones (figure S1) using the Tropic of Capricorn and southern and northern polar circles as boundaries: the north frigid, north temperate, tropical, south frigid, and south temperate zones (figure S1). These zones largely reflect the basic horizontal global climate distribution.

Methods
Based on the C-C thermodynamic equation, the nonlinear relationship between saturated water vapour pressure (e sat ) and air temperature (T) can be described as follows (Koutsoyiannis 2012): where e s0 is the saturation water vapour pressure (=611 Pa) at temperature T 0 (=273.16 K), L v is the latent heat of vaporisation (=2.5 × 106 J kg −1 ), and R v is the water vapour gas constant (=461 J kg −1 K −1 ). Accordingly, the differential expression of the C-C relation is as follows: 'Binning scaling' , which is used to identify nonlinear relationships among variables, has been extensively applied to analyse climate responses to global warming over the past decade (Prein et al 2017, Sullivan et al 2020, Fowler et al 2021). We applied a binning scaling method described previously (Lenderink and Van Meijgaard 2008, Lenderink et al 2011, Drobinski et al 2016 to calculate the scaling rate between precipitation extremes and temperature. For each 1.0 • × 1.0 • grid, precipitation intensity values >0.1 mm d −1 were defined as wet events, and same-day temperatures were ranked. The ranked temperature-precipitation pairs were allocated to 12 bins; the median daily temperature represents the local temperature of each bin, and precipitation intensity values above the 99th quantile (Koenker and Bassett 1978) represent extreme events. Some studies have demonstrated that using equal temperature intervals to divide datasets can cause empty bins (Wasko and Sharma 2014), leading to skewed relationships between precipitation and temperature (Herath et al 2018). Therefore, temperature ranges were permitted to differ among bins in this study, to guarantee an equal sample size for each bin (Hardwick Jones et al 2010, Drobinski et al 2018. Atmospheric water vapour pressure increases non-linearly with temperature; at 25 • C, e sat increases at a rate of approximately 6.8% • C −1 . Because there is a similar exponential relationship between precipitation extremes and temperature, the following regression equation linking precipitation to temperature was established to quantify the response of precipitation extremes to global warming (Zeder and Fischer 2020): where P 1 and P 2 are the intensity values for precipitation extremes in adjacent bins, and α P is the scaling rate of precipitation extremes.
Climate responses show monotonically increasing, monotonically decreasing, and hook structures with precipitation extremes. The climate response can be identified from the fitted line of the P-T relationship. The peak point temperature (defined as the inflexion point of the fitted line, T pp ) is determined by locally weighted scatterplot smoothing (Cleveland 1979). The scaling rate is inferred using least-squares linear regression (Yin et al 2018). For hook-shaped P-T curves, the regression line is fitted only up to the peak temperature (Mishra et al 2012, Miao et al 2016. DPT is the temperature at which the air cools to the point of water vapour saturation under constant water vapour content and air pressure. The ERA5 2 m air temperature (T 2m ) and DPT (T dew ) data were substituted into equation (1) to derive near-ground RH, as follows: RH = esat(T dew ) esat(T2m) . To account for the influence of thermal and dynamic factors on precipitation extremes, TCWV, CAPE, and RH data for the day of each precipitation extremes event in each bin were selected and matched with the DPT, using scaling rates derived from equation (3). Figures 1 and 2 show density scatter plots of the 99th percentiles of precipitation and their corresponding temperatures for each grid within five study areas (figure S1). Precipitation show a monotonic increase in intensity with SAT in the north and south Relationship between 99th percentile precipitation and near-surface air temperature (SAT) in each temperature zone, based on (a) 1-hourly, (b) 3-hourly, (c) 6-hourly, and (d) daily peak precipitation intensity data. Temperature zones include the (1) north frigid, (2) north temperate, (3) tropical, (4) south temperate, and (5) south frigid zones. Precipitation data are on a logarithmic scale. Black dashed lines indicate Clausius-Clapeyron (C-C) scaling. frigid zones. In the north and south temperate zones, there is a pronounced hook structure between SAT and precipitation, with peak point temperatures of ∼20 • C-30 • C (figure 1). This hook structure indicates a change from a positive to negative correlation. In figure 2, in the tropics, higher slopes appear in the P-T relationship, and in the southern temperate zone, some of P-T curves also show steep slopes, indicating 'super C-C' scaling. However, the coarse zonal distribution data does not allow detailed analysis of the climate distribution. P-T relationships are similar between the north and south frigid regions according to both the SAT-(figure 1) and DPT-based results (figure 2). However, unlike the SAT results, there is no significant hook structure between precipitation and temperature in temperate regions according to the DPT results, suggesting that they are closer to the expected C-C relationship. The fitted line of the precipitation-DPT relationship for the tropics show no negative correlation, but the slope is steeper than for other regions. As the overall precipitation and temperature data are stacked, it is unclear whether hook structures are present in tropical regions. Although the scaling pattern of precipitation extremes and temperatures show variations under different durations, the uncertainty is far smaller than the spatial heterogeneity. The relationship between temperature and precipitation is insensitive to timescale (figures 1 and 2). The general trends in each study area (figure 2) are poorly represented at these spatial scales; therefore, we further analyse the response of precipitation to temperature at the grid scale.

Scaling pattern and hook structure
On a global scale, there are three types of P-T curve (figures 3 and 4), i.e. monotonically increasing, monotonically decreasing, and hook structure . The hook structure reflects increasing precipitation intensity with temperature up to a certain threshold (i.e. T pp ), followed by a decrease in precipitation intensity as the temperature continues to increase. The spatial distributions of the precipitation-SAT relationship and T pp are shown in figures 3(a) and (b), and that for the precipitation-DPT relationship is shown in the remaining subplots (figures 3(c)-(f)). Except for the Antarctic and Arctic circles, T pp increases from frigid regions to the Equator; it ranges from 15 • C to 25 • C in most regions, with a monotonic increase seen in the P-T relationship in most 30-60 • S and 30-60 • N regions. Although the overall distribution of the three types of P-T curve and T pp trend are similar between SAT and DPT, there are large differences in tropical regions. The frequency of hook structures varies, with monotonic increases seen in approximately 40% and 60% of regions worldwide for SAT and DPT, respectively; the proportion of regions showing monotonic decreases declining from 10% to approximately 1%. Compared to SAT, fewer regions show a hook structure between precipitation extremes and DPT (38% and 31% of all regions, respectively), and monotonically decreasing relationships are largely absent in the tropics. According to both the SAT-and DPT-based results, the proportion of monotonically increasing patterns decreases as time resolution of precipitation increases, whereas the proportions of monotonically decreasing and hook structure increases. For the DPT, the percentage of hook structures increased gradually from 26% to 31%, while that of monotonically increasing patterns decreased from 65% to 59%, as the precipitation duration increased from 1 h to 1 d.  Figure 5 shows the scaling rate distribution of 99th percentile hourly and daily precipitation intensity values based on SAT. Approximately 10% of all grids (mainly in humid and tropical regions) shows a negative correlation between precipitation and SAT, and about 70% of all grids with a scaling rate no more than twice the C-C rate (approximately 13.6% • C −1 ). Except for the tropics, global scaling rates generally decrease with increasing latitude. Scaling rates of 5% • C −1 -10% • C −1 (similar to the C-C rate) are observed over most northern temperate terrestrial and marine areas, whereas rates exceeding 10% • C −1 dominate southern temperate marine areas (figure 4). Large numbers of negative and ultrahigh scaling rates (more than twice the C-C rate) are observed in the tropics, consistent with previous studies (O'Gorman 2012, Wang et al 2017). The proportion of grids with negative scaling rates differs between South America and Africa according to the hourly and daily data.

Scaling rates of precipitation events by duration
Scaling rates decrease slightly from the hourly to daily scale in many regions, such as Australia, southern temperate marine areas, and North America, suggesting that the scaling rate is influenced by precipitation duration, and that the effect of climate on the likelihood of short extreme precipitation events may differ from that predicted by the C-C relationship. As moisture availability is the main factor constraining extreme precipitation events at higher temperatures, we use DPT as the temperature variable to analyse the effect of moisture on scaling rates at the hourly, 3-hourly, 6-hourly, and daily scales.
Scaling rates for 99th percentile precipitation intensity values based on DPT at the global scale are shown in figure 6. The overall trend is similar to that based on SAT. Globally, >60% of the scaling rates were in the range of 0% • C −1 -10% • C −1 . However, in the tropics, very few grids are negative scaling rates, and the proportion of scaling rates more than twice the C-C rate increases by approximately 10%. Comparison of hourly, 3-hourly, 6-hourly, and daily datasets shows that scaling rates decrease with increasing precipitation duration in Oceania, South America, Africa, and southern temperate marine areas; all other regions show smaller changes in scaling rates as the precipitation duration increased. Except in the tropics, most SAT results are smaller  than DPT results in terrestrial areas, while vice versa in marine areas ( figure 7). Also, the difference in scaling rates between temperatures is narrower in terrestrial areas than these in marine areas.
To further compare the impact of type of temperature data and precipitation duration on median scaling rates at different latitudes, we generate box plots ( figure 8). The median scaling rates trends are generally consistent between the two types of temperature data; they increase with decreasing latitude, except in the tropics, generally followed the C-C scaling trend in north and south temperate regions, and are slightly lower than the C-C scaling values in southern and northern boreal zones.
The scaling rates differ considerably between the two types of temperature data in the tropics. Box plots  of scaling rates for the precipitation duration-DPT relationship are shown in figure 9. Whereas the scaling rates for SAT range widely from positive to negative, those for DPT generally exceed the C-C scaling values. A negative correlation between precipitation and DPT is observed in some regions near the Equator. The precipitation period also modestly affects scaling rates, which declined slightly as precipitation duration increased; this finding is somewhat at variance with the sharper decrease observed in a previous study (Visser et al 2021). We suggest that more significant differences in scaling rate between the two types of temperature data may be revealed with individual precipitation event based analyses. Thus, higher-resolution (sub-hourly scale) data is needed for deeper understanding the P-T relationship, although such data are more difficult to obtain.  Dashed blue and red lines are the C-C rate and C-C rate doubled, representing scaling rates of 6.8% • C −1 and 13.6% • C −1 , respectively. Figure 10 shows scaling rates of CAPE, RH, and TCWV for DPT on days with extreme precipitation events. CAPE is commonly used to assess atmospheric instability and potential for convective strength. Except for the 90 • N to 60 • N area, the scaling rates of CAPE are much higher than 6.8% • C −1 ( figure 10(a)), suggesting that the thermodynamic effects of atmospheric pose a positive contribution on the intensification of precipitation extremes. Otherwise, unlike scaling rates of P-T relations, there are no apparent regional differences in CAPE results.

Scaling of meteorological factors with DPT
Because atmospheric water vapour pressure is determined by both saturated water pressure and RH. As temperatures increase, e sat scales at the rate of 6.8% • C −1 , whereas that of RH is ∼0% • C −1 ( figure 10(b)), which suppresses the response of TCWV to temperature. The RH scaling rates for terrestrial areas is less than for marine areas, except for some land areas in the tropics. The scaling between global TCWV and DPT is positive and at the rate above 6.8% • C −1 (figure 10(c)), the trend for total precipitable water is expected to continue to increase in the future. However, TCWV scaling rates are much lower than those of precipitation extremes in tropical regions, potentially leading to hook structures in those regions. Figure 10. Scaling rates of (a) convective available potential energy (CAPE), (b) relative humidity (RH), and (c) total column water vapour (TCWV) for DPT on days with extreme precipitation events. Red shading indicates the scaling rate range; solid dark blue line indicates the median scaling rate; dotted line indicates C-C scaling.

Discussion and conclusion
The results of this study suggest that the peak of the P-T curve varies among climate zones. The T pp threshold is relatively symmetrical across latitudes, ranging from 20 • C to 25 • C in the tropics and 10 • C and 20 • C in northern and southern temperate zones. DPT-based scaling rates are mainly positive and had a more homogenous spatial distribution than the SAT-based scaling ones, which show large numbers of negative values in the tropics. This constraint imposed by RH accounts for the high frequency of negative scaling rates for SAT in the tropics, which are rare for DPT. Negative skew is observed in the SAT data distribution, which reduces the odds of sampling a cold day in the coldest bin during wet events; this can lead to large discrepancies between climate and wet events in cold bins, and thus to negative correlations between precipitation and SAT.
Despite the dominance of positive scaling rates for DPT, a hook-shaped P-T relationship is still evident. Moreover, the super C-C scaling rates also exhibit in many coastal areas of the tropics. This hook structure is a global phenomenon that can result from interactions among several factors. Warming has weakened circulation, thus inhibiting convection (O'Gorman et al 2018). Fluctuations in vertical wind speed prevent convection and reduce the supply of surrounding water vapour (Pendergrass and Gerber 2016). The reduction of water vapour transport capacity impedes precipitation processes, ultimately resulting in a hook structure between precipitation extremes and climate. In marine areas, the precipitation response to temperature is not fully understood. Sub-C-C scaling values may be caused by applying the traditional approach of linking precipitation extremes to mean temperature, where some studies have demonstrated that local mean temperature increases exceed peak temperatures over land (Wang et al 2017). Precipitation extremes vary between datasets, which can also lead to deviations from the C-C scaling rate (Hosseini-Moghari et al 2022).
In the tropics, the hook structure is more prominent, which may be linked the drier and hotter conditions; a decrease in RH can lead to a substantial increase in land temperature (Byrne 2021, Yin et al 2023b. The more obvious hook structure and super C-C scaling rates observed in the tropics in this study are also related to precipitation duration, where the proportion of regions exhibiting hook structures increased, while the scaling rate decreased, with extended rainfall. This can be explained by the occurrence of precipitation extremes over a narrow range of high temperatures. Most convection-dominated precipitation events are short-lived, and are less frequent under the high-temperature conditions (Visser et al 2021) in the tropics.
The ERA5 data is further analysed in terms of variations in TCWV, CAPE, and RH with increasing temperature during extreme precipitation events. CAPE is an important indicator of thermal atmospheric parameters, the maximum potential energy available to air parcels for convection (Lee and Byun 2011). Higher CAPE values indicate a higher probability of severe storm process. In addition, the change of extreme precipitation in marine and terrestrial regions show different characteristics, while the diversity of CAPE scaling rates in different regions are not obvious, implying that thermodynamic response mechanisms of sea and land may differ. CAPE increases during extreme precipitation and vertical wind speeds are enhanced. The upwelling provides a large amount of water vapour to the atmosphere, which facilitates the enhancement of the humidity factor including RH and TCWV. The increase in atmospheric water vapour content further promotes convection and the formation of extreme precipitation.
Our results indicate that water vapour and precipitation respond differently to global warming. The global distribution of precipitation extremes and TCWV scaling rates indicated that precipitation extremes frequently occur along the tropics and over wetter land regions; higher TCWV scaling rates were distributed along the tropics. Kim et al (2022) found that there is a strong correlation between precipitable water and temperature in the tropics land, and that increasing precipitable water lead to increasing extreme precipitation events with shorter duration. Super-C-C rates in tropics regions are thus associated with positive scaling rates of TWVC. TCWV scaling rates are much lower than those of precipitation extremes in tropical regions, consistent with previous studies showing that atmospheric moisture is not the only factor driving precipitation extremes (Zhang et al 2019a, Wang andSun 2022), and that precipitation extremes increase more rapidly than mean precipitation (Pendergrass and Gerber 2016). The tendency for RH to increase with warmer temperatures is weaker than that of C-C. Moreover, TCWV is limited by atmospheric moisture, resulting in a declining trend of precipitation extremes at higher temperatures, suggesting that the hook structure is mainly caused by dynamic constraints.
The wide variation in scaling rate according to region, and the hook structure, seen herein indicate that using the C-C relationship to infer precipitation extremes under global warming remains problematic (Zhang et al 2017). Moreover, variation in P-T relationships lead to uncertainties when analysing precipitation extremes according to temperature. Therefore, the scaling rate of the P-T relationship cannot be considered in isolation; other climatic factors involved in thermodynamic and kinetic processes must be taken into account. There remains considerable inconsistency in reported precipitation extremestemperature relationships in the tropics; seasonal variations in temperature, water vapour sources, the influence of the ocean, and atmospheric circulation must be considered to reduce uncertainty and elucidate the mechanisms driving extreme tropical precipitation.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). the Central Universities (No. 2042022kf1221). This work is also supported by the Open Research Fund of Hubei Key Laboratory of Intelligent Yangtze and Hydroelectric Science, China Yangtze Power Co., Ltd. The numerical calculations in this paper have been performed on the supercomputing system in the Supercomputing Centre of Wuhan University. ERA5 reanalysis data are downloaded from the Copernicus Climate Change Service (C3S) Climate Data Store (https://cds.climate.copernicus.eu/). The results contain modified Copernicus Climate Change Service information 2020. Neither the European Commission nor ECMWF is responsible for any use that may be made of the Copernicus information or data it contains.